Black hole merger estimates in Einstein-Maxwell

and Einstein-Maxwell-dilaton gravity

Puttarak Jai-akson;a Auttakit Chatrabhuti,a Oleg Evnin,a,b Luis Lehnerc

a Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, Thailand

b Theoretische Natuurkunde, Vrije Universiteit Brussel and International Solvay Institutes, Brussels, Belgium

c Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J 2W9, Canada

puttarak.jaiakson@gmail.com, auttakit@sc.chula.ac.th, oleg.evnin@gmail.com, llehner@perimeterinstitute.ca

ABSTRACT

The recent birth of gravitational wave astronomy invites a new generation of precision tests of general relativity.Signatures of black hole (BH) mergers must be systematically explored in a wide spectrum of modified gravity theories.Here, we turn to one such theory in which the initial value problem for BH mergers is well posed, the Einstein-Maxwell-dilaton system. We present conservative estimates for the merger parameters (final spins, quasinormal modes) basedon techniques that have worked well for ordinary gravity mergers and utilize information extracted from test particlemotion in the final BH metric. The computation is developed in parallel for the modified gravity BHs (we specificallyfocus on the Kaluza-Klein value of the dilaton coupling, for which analytic BH solutions are known) and ordinary Kerr-Newman BHs. We comment on the possibility of obtaining final BHs with spins consistent with current observations.

arX

iv:1

706.

0651

9v2

[gr

-qc]

17

Aug

201

7

2

I. INTRODUCTION

The spectacular detections of GW150914 [1], GW151226 [2] and GW170104 [3] resoundingly marked the beginningof gravitational wave astronomy. The new observational window opened by such a feat is offering unprecedentedopportunities to scrutinize our Universe and probe fundamental questions. Among these, perhaps the most excitingprospect is to examine gravity in highly dynamical/strongly nonlinear regimes for the first time, and to put generalRelativity (GR) through the most stringent tests to date. The abovementioned signals, produced by merging binaryblack holes (BHs), have been shown to be consistent with GR [4, 5]. Deeper scrutiny will be gradually possible in thecoming years (e.g. [6–8]) as more events and higher signal-to-noise is achieved in binary BH detections. (Even furthercomplementary tests will be made possible when nonvacuum binaries are detected. This will discriminate betweentheories giving rise to the same dynamics as in GR in binary BHs systems, but producing nontrivial differences whenat least one neutron star is involved [9–12].)

Importantly, with the information so far available (and GR remaining consistent with observations), it is naturalto expect that any deviations from GR will be subtle. This implies that the search for potential deviations is adelicate task, especially given the fact that signals will be typically buried in the aLIGO/VIRGO noise.1 To facilitatethis task, theoretical guidance is required for detection and analysis. Such guidance is gradually becoming availablethrough phenomenological approaches [13, 14], or through explicit calculations of merger dynamics within possibleextensions to GR [9–11, 15]. While the former makes minimal assumptions with respect to such extensions, thelatter requires understanding the complex nonlinear behavior of modified gravities. This, in turn, can only be donewithin mathematically well-defined theories [16] (see also, e.g. [17, 18]). (Most extensions/alternatives to GR arenot formulated in a way leading to a well-posed problem due to the presence of higher derivatives, ghosts, a suspectinitial value problem, etc. Incipient work is exploring how to handle these otherwise reasonably motivated theories,e.g. [19–21])

With data coming in at an increased rate in the immediate future, from a theoretical point of view, it is imperativeto provide a sound guidance covering a range of relevant theories. The principal target for this type of analysis is toidentify the key signatures of the waveforms (during the transition from inspiral to plunge, and in postmerger behavior)which would provide important insights into the dynamics of the system as well as the nature of the objects involvedin the merger event.

In the present work, we take a step toward the comprehensive analysis of modified gravity mergers, focusing onthe particular framework of the so-called “Einstein-Maxwell-dilaton” (EMD) theory. In this theory, in addition to thestandard tensor (metric) field, a scalar and a gauge field are present. The presence of a gauge field allows, in particular,for the BH to sustain nontrivial hair and the system to radiate scalar and vector modes. This theory is motivated byvarious low-energy limits of string theories, and is thus a natural candidate to explore deviations from standard gravity.In the EMD theory, explicit analytic solutions called the KK BHs are known for a specific value of the dilaton couplingparameter. While we expect that the behaviors are qualitatively similar at different comparable values of the dilatoncoupling parameter, in our derivations, we focus on this value that makes the situation analytically tractable. Wefurthermore systematically compare our derivations with the corresponding results in the standard Einstein-Maxwelltheory.

Full numerical simulations of gravitational systems are very costly, in standard GR and, even more so, with additionalfields present (see, e.g. [22, 23]). It is important to identify not fully rigorous but reliable estimates for the mergerprocesses, which would precede and guide costly numerical work. In ordinary GR, it has been rather solidly establishedthat information on test particle motion in the final state BH can be utilized to build estimates for the merger dynamicswith a precision on the scale of 10%. Thus, the analysis of so-called “innermost stable circular orbits” (ISCO) formassive particles can produce accurate estimates for the final spin of the merger via what is referred to as the BKLrecipe, after the initials of the authors of [24] where it was introduced. The circular orbits for massless particles, knownas the “light ring”, provide information on the quasinormal modes of the final BH, and therefore gravitational waveemission patterns at the late stages of a relaxation of the merger product, the “ring down” [25] (see however [25, 26]for limitations).

We see the type of estimates we present here as a first step in two significant directions. First, the results can guidefuture numerical simulations of BH collisions in the EMD theory [27] (simulations of collisions of Reissner-NordströmBHs involving Maxwell fields have been previously reported in [28–30]). Second, the type of estimates we presenthere are straightforwardly applicable in other modified gravity theories in which explicit BH solutions are known. For

1 Future facilities like the space-based LISA, and planned ET and Cosmic Explorer will have a much higher sensitivity though they areover a decade away. Nevertheless, coherent analysis of multiple events in aLIGO/VIRGO can boost SNR by a significant amount toextract subtle features of the signal, e.g. [8].

3

example, an analytic treatment of geodesics in STU BH spacetimes that generalize KK BHs has just appeared in [31].(Full numerical simulations in generic modified gravity theories would have to be preceded by in-depth analysis of thecorresponding equations of motion to ascertain that the collision problem is well posed.)

Our estimates of final merger spins invite some contemplation of the potential “low spin issue” of individual blackholes involved in the merger. The LIGO detections point to spin-to-mass ratio of the BH resulting from the mergerbeing quite close to what would have resulted from colliding binary BHs with intriguingly small spins, if these werealigned with the orbital angular momentum. Alternatively, such scenario also arises from spin configurations with arather small projection of their spin along the direction of the orbital angular momentum – a puzzling possibility onastrophysical grounds. The estimates for charged BHs we present here make it possible to lower the final spin of themerger at generic values of the collision parameters. While charged BHs are not part of the standard astrophysical lore,they have occasionally been evoked in addressing possible observational paradoxes (see [30] and references therein).

The paper is organized as follows. In Sec. II, we review the background material on our estimation techniques andthe BH metrics involved. In Sec. III, we show how the original BKL recipe utilizing pure geodesic motion can beapplied to Kaluza-Klein BHs in the EMD theory. In Sec. IV, we incorporate corrections to the test particle motiondue to the presence of charges and develop improved estimates. In Sec. V, we repeat these derivations for the standardKerr-Newmann BHs of ordinary gravity, and in Sec. VI compare these results with what we have obtained in modifiedgravity. We finally provide a summary in Sec. VII.

II. GENERALITIES

A. Final spin estimation: The BKL recipe

Our strategy is simple and relies on “conservation arguments” to estimate the final BH mass and angular momentumresulting from quasicircular binary BH mergers as presented in [24] (often referred to as the BKL approach). Onethinks of the initial phase of the merger process, in the low eccentricity case, as a gradual contraction of the binaryorbit due to the energy loss via a gravitational wave emission. This phase cannot proceed indefinitely however, sincecircular orbits become unstable once the two BHs get closer than a certain distance apart. (This distance is knownas the ISCO radius.) Once this moment has been reached, a ‘plunge’ occurs resulting in the final BH formation.Since during the plunge only a small amount of angular momentum is radiated, one can use the angular momentumconservation and the information on the ISCO to estimate the final BH spin.

The BKL approach can be viewed as an extrapolation of the test-particle (extreme-mass-ratio) behavior to thecomparable mass case [24]. That such an approximation is able to capture the correct behavior even in the equal massregime follows naturally from regarding the merger as described perturbatively with respect to the final BH spacetime.Both theoretical studies and the behavior inferred from recent gravitational wave observations with LIGO [1, 2] lendsupport for such a picture.

For simplicity, we assume the change in masses is small and thus estimate Mfinal = M1 +M2 (Further improvementscan be incorporated as in [32], but the resulting differences are small so this assumption is adequate for our currentpurposes). Conservation of angular momentum at the moment of plunge implies [24],

MAf = Lorb(rISCO, Af ) +M1A1 +M2A2, (2.1)

where M1,M2 are the initial masses of BHs, and M = M1 +M2 is the mass of the merger product BH. A1 and A2 arethe initial spin parameters, Af is the final BH spin. Lorb(r,Af ) is the angular momentum of a test particle carryingthe reduced mass µ = M1M2/M orbiting around the final BH of mass M and spin parameter Af on a circular orbitof radius r, and rISCO is the radius of the ISCO. We will assume that the angular momentum of each individual BH iseither aligned or counteraligned with respect to the orbital angular momentum (misalignments can be accounted forby suitable projections as explained in [24]).

For future use, it is convenient to reexpress the above equation for Af through χi = Ai/Mi and ν = M1M2/M2 as

Af = l(rISCO, Af )ν +Mχ1

4(1 +

√1− 4ν)2 + Mχ2

4(1−

√1− 4ν)2. (2.2)

where l(r,Af ) refers to the angular momentum of a unit mass test particle on a circular orbit. Both rISCO andl(rISCO, Af ) are completely expressible through geodesic motion in the metric of the final BH. Equation (2.2) is solvedto obtain an estimate for the final BH spin Af . In this work, we will apply this technique to the case of BHs in theEinstein-Maxell-dilaton and Einstein-Maxwell theories.

4

B. The light ring

As we have just explained, ISCO analysis for massive test particles allows for the estimation of the final spin of themerger via the BKL recipe. Additional information on the merger process can be extracted by considering lightlikeorbits in the final BH metric.

At the final stages of BHs mergers, the merger product settles to a stationary configuration, which is known as theringdown stage. This stage is primarily characterized by linearized vibrational modes with complex-valued frequencies,known as the quasinormal modes (QNMs), in the background of the final BH. The frequencies of BH quasinormalmodes can be effectively approximated by considering unstable geodesics of massless particles, also known as the lightring [33] (see however [25, 26] for a discussion of subtleties). The QNM frequency can be estimated along these linesas

ωQNM = Ωcj − i(n+1

2)|λ|, (2.3)

where n is the overtone number and j is the angular momentum of the perturbation. The real part of QNM frequenciesis determined by the angular velocity at the unstable null geodesic Ωc, and the imaginary part, λ denotes the Lyapunovexponent, which is related to the instability time scale of the orbit. The radial equation of motion for a massless testparticle can be generically written in the form

ṙ2 = Veff(r). (2.4)

The Lyapunov exponent can be computed as

λ =

√V ′′eff2ṫ2

, (2.5)

with this expression evaluated at the unstable null geodesic. We shall demonstrate how this evaluation works inpractice in subsequent sections. At this point we find it important to stress that it is not known whether the lightring calculation produces reasonably accurate estimates of the QNMs in generic extensions to GR. For the EMD casewe focus in this work, further support for this approach is provided by: (i) Recent studies in full nonlinear regimeswhich not only illustrates the QNM behavior but also stresses how BHs in this theory can be regarded as interpolatingbetween charged to neutral black holes in GR when considering small to large values of the dilaton coupling. (ii)Calculations of QNMs and direct comparisons with results from the light-ring calculation presented in [34]. Of course,a rigorous treatment requires the calculation of QNMs through a linearized study but given the dearth of such studiesfor the (many) extensions to GR in existence, our approach provides a rather simple way to build intuition (seealso [35]).

C. Einstein-Maxwell-dilaton BHs

The approach discussed above relies on understanding the behavior of test particles in suitable BH spacetimes. Toexplore mergers in an extension to general relativity, we consider here the case of the Einstein-Maxwell-dilaton theorywhich arises as a low energy limit in string theory. The action of this theory is given by [36],

S =

∫d4x√−g[−R+ 2(∇Φ)2 + e−2αΦF 2]. (2.6)

For charged rotating BHs, analytic solutions (Kaluza-Klein BHs) are only available for the dilaton coupling α =√3 [37, 38], known as the Kaluza-Klein (KK) value of the coupling.2 We shall hereafter focus on these particular

solutions, though we do not anticipate dramatic differences for other values of the coupling.The metric for the KK solution in spherical coordinates is

ds2 = −1− ZB

dt2 − 2aZ sin2 θ

B√

1− v2dtdφ+

[B(r2 + a2) + a2 sin2 θ

Z

B

]sin2 θdφ2 +BΣ

(dr2

∆+ dθ2

), (2.7)

2 Numerical solutions describing the behavior of single and binary BH systems for a broad set of α values will be presented in [27].Importantly for our discussions, such black holes appear to be stable and the black hole mergers behave qualitatively similar to the onesobtained in GR.

5

where

B =

(1 +

v2Z

1− v2

)1/2, Z =

2mr

Σ, ∆ = r2 + a2 − 2mr, and Σ = r2 + a2 cos2 θ. (2.8)

The vector potential and the dilaton field are

At =v

2(1− v2)Z

B2, Aφ = −a sin2 θ

√1− v2At, and Φ = −

√3

2lnB. (2.9)

The physical mass M , charge Q, and angular momentum J are expressed through m, v and a as

M = m

(1 +

v2

2(1− v2)

), Q =

mv

1− v2, and J =

ma√1− v2

. (2.10)

(One may recognize boostlike dependences on v, and indeed, four-dimensional Kaluza-Klein BHs descend from boostedBH solutions in five-dimensional gravity.)

For completeness, we also quote the standard Kerr-Newman metric for a charged rotating BH in ordinary generalrelativity. For a BH of mass M , spin a, and electric charge Q in (t, r, θ, φ) coordinates, this metric has the form

ds2 =−(

1− 2Mr −Q2

ρ2

)dt2 − 2(2Mr −Q

2)a sin2 θ

ρ2dtdφ+

ρ2

∆dr2 + ρ2dθ2 +

sin2 θ

ρ2((r2 + a2)2 − a2∆ sin2 θ)dφ2,

(2.11)

where

∆ = r2 + a2 − 2Mr +Q2, and ρ = r2 + a2 cos2 θ. (2.12)

The corresponding vector potential is

At =Qr

ρ2, Aφ = −

Qar sin2 θ

ρ2. (2.13)

D. Newtonian limit of charged particle motion

In the standard BKL recipe [24], one relies on pure geodesic motion, and therefore the mass of the test particle doesnot affect the shape of its trajectory, or the location of the ISCO. Once electromagnetic effects are taken into account,the motion of test particle depends on its charge-to-mass ratio. We shall now briefly examine the Newtonian limit ofthe test particle motion and identify reasonable mass and charge assignments for our generalization of the BKL recipe.

The motion of a test particle of mass µ and charge q is described by the action,

L = 12µgλν ẋ

λẋν − qAν ẋν , (2.14)

and the corresponding equation of motion

µ

(ẍµ + Γµνρẋ

ν ẋρ)

= −qẋνFµν . (2.15)

One has to be careful choosing the sign in front of q in the action. We shall see below that our choice of the sign, incombination with the standard parametrization of BH solutions, reproduces the correct Coulomb force for motion oftest particles in the Newtonian limit.

To reproduce the Newtonian limit, we impose ẋi � ṫ; m, a� r. The above equation of motion reduces to

µ

(ẍµ + Γµ00ṫ

2

)= −qṫFµ0. (2.16)

6

For the sake of parameter identification, we specialize to purely radial motion. For the metric and field strengthcorresponding to the KK BH solution, one gets

µ

(d2r

dt2+m

r2

)=

q

r2mv

1− v2. (2.17)

At q = 0, this obviously reproduces the classical equation of motion in Newtonian gravity. Assuming v � 1 (which isequivalent to Q�M) and expressing everything through the physical mass and charge of the BH given by (2.10), werecover a radial motion equation due to Newtonian gravity and Coulomb force (note the correct sign of the Coulombterm),

µ

(d2r

dt2+M

r2

)=qQ

r2. (2.18)

This can be compared to the dynamics of two particles of masses M1 and M2, and charges Q1 and Q2 governed bythe equation of motion,

M1M2M1 +M2

d2r

dt2+M1M2r2

=Q1Q2r2

(2.19)

One of the ingredients of the BKL recipe is to approximate the motion of BHs during an approach by the motion inthe metric of the final BH. If we assume that the final BH has a mass M = M1 + M2, and a charge Q = Q1 + Q2,guided by the above Newtonian limit, it is reasonable to assign the following mass µ and charge q to the test particle,which makes (2.18) and (2.19) agree:

µ =M1M2M1 +M2

, q =Q1Q2Q1 +Q2

. (2.20)

Remark: Ordinary charged rotating BHs in general relativity are described by the Kerr-Newman metric (2.11). Themotion of a charged particle around the Kerr-Newman BH in the Newtonian limit (2.16) is identical to (2.18). Thus,by comparing with the Newtonian equations (2.19), we recover the parameters of the test particle (2.20) in the BKLrecipe. Note that for the Kerr-Newman case, we do not need to impose the small charge condition Q�M .

We are now ready to apply the BKL approach to estimate the outcome of binary BH mergers in the EMD theory.We organize this computation by first neglecting the effect of charges on test particle motion (but retaining it in themetric). This estimate based on pure geodesic motion is directly inherited from the original BKL considerations, andit immediately applies if one of the colliding binaries has a negligible charge. The estimates based on pure geodesicsare also technically simpler and produce reasonable results even in the presence of charges, as we shall eventually see.After completing the derivation based on pure geodesic motion, we turn to more accurate estimates incorporating theeffects of charges.

III. MERGER ESTIMATES FOR KK BHS BASED ON PURE GEODESIC MOTION

A. Kinematic considerations

1. Orbits in the equatorial plane

Consider the motion of a neutral test particle in the equatorial plane of a KK BH (2.7), forced by the conditions

θ = π/2 and θ̇ = 0. The relevant metric components are,

gtt = −1

B(1− 2m

r), gtφ = −

2γma

rB, grr =

Br2

∆, gφφ = B(r

2 + a2) +2ma2

rB, (3.1)

where B2 = 1 + 2m(γ2 − 1)/r. We will work with positive final BH charges Q corresponding to v > 0 (this is amatter of convention as the sign can always be flipped), and replace the boost parameter v in the metric (2.7) withthe ‘Lorentz factor,’

γ ≡ 1√1− v2

≥ 1. (3.2)

7

In our derivations, we shall repeatedly use the identity

g2tφ − gttgφφ = ∆. (3.3)

The Lagrangian of a unit mass test particle is given by,

L = gttṫ2 + 2gtφṫφ̇+ gφφφ̇2 + grr ṙ2, (3.4)

where dots represent derivatives with respect to the proper time τ . Because none of the metric components dependon t or φ, the corresponding conjugate momenta, which are just the total energy ε, and the angular momentum l ofthe test particle, are conserved,

ε = −gttṫ− gtφφ̇, l = gtφṫ+ gφφφ̇. (3.5)

One therefore has

ṫ =gtφl + gφφε

∆, φ̇ = −gttl + gtφε

∆. (3.6)

We now focus on circular orbits (ṙ = 0). The equations of motion are given by

gttṫ2 + 2gtφṫφ̇+ gφφφ̇

2 = −1 (3.7)g′ttṫ

2 + 2g′tφṫφ̇+ g′φφφ̇

2 = 0 (3.8)

where primes denote derivatives with respect to the radial coordinate r. The second equation is the r-component ofthe Lagrangian equations of motion, while the first one enforces τ to be the proper time. Using (3.7), (3.5) and (3.3),we write ε as

ε2 = φ̇2∆− gtt. (3.9)

The expression for φ̇ can be found by solving (3.7) and (3.8) simultaneously:

φ̇ = ± g′tt(

g′tt(2gtφg′tφ + gttg

′φφ)− gφφ(g′tt)2 − 2gtt(g′tφ)2 ± 2(gtφg′tt − gttg′tφ)

√(g′tφ)

2 − g′ttg′φφ

)1/2 . (3.10)The upper sign refers to prograde orbits, and the lower sign to retrograde orbits. After evaluating the derivatives, wehave the expression

φ̇ = ± V (m2/4rU)

14(

(2m− r)W − rV (mV + (m− r)U)± 2aγ√mrU [rV (V + (m− r)(γ2 − 1))−W ]

)1/2 , (3.11)where

U = r + 2m(γ2 − 1),V = r − 2m+ (r + 2m)γ2,W = ma2(γ2 − 1)2. (3.12)

Knowing ε and φ̇, one can use (3.6) to find the value of l corresponding to the given circular orbit as

l = − 1gtt

(φ̇∆ + gtφε

). (3.13)

Note that the expressions for ε and l presented in (3.9) and (3.13) reduce to the expressions for the Kerr BH [39] whenγ = 1, i.e., in the absence of charge.

8

2. The ISCO

For a massive particle moving on a general trajectory in the equatorial plane, we have

− 1 = gttṫ2 + 2gtφṫφ̇+ gφφφ̇2 + grr ṙ2. (3.14)

Writing the above equation in terms of conserved quantities ε and l using (3.6) and (3.3), one gets

ṙ2 +1

Br2(−gttl2 − 2gtφlε− gφφε2 + ∆) = 0. (3.15)

We now define the effective potential as

Veff ≡1

Br2(gttl

2 + 2gtφlε+ gφφε2 −∆). (3.16)

In order to find the ISCO radius, we have to impose the conditions (see [39] for the Kerr BH),

Veff = 0,d

drVeff = 0, and

d2

dr2Veff = 0. (3.17)

The first two conditions simply enforce circularity of the orbit, and could be equivalently replaced by constraints onconserved quantities of circular orbits from the previous section. It is, however, convenient to deal with the aboveformulation in terms of the effective potential, which results in the following three equations:

gttl2 + 2gtφlε+ gφφε

2 = ∆,

g′ttl2 + 2g′tφlε+ g

′φφε

2 = ∆′,

g′′ttl2 + 2g′′tφlε+ g

′′φφε

2 = ∆′′. (3.18)

Solving the above equations for ε2, we get

ε2 =g′tφg

′′tt∆− gtφg′′tt∆′ − g′ttg′′tφ∆ + gttg′′tφ∆′ + gtφg′tt∆′′ − gttg′tφ∆′′

gφφg′tφg′′tt − gtφg′φφg′′tt − gφφg′ttg′′tφ + gttg′φφg′′tφ + gtφg′ttg′′φφ − gttg′tφg′′φφ

. (3.19)

Substituting ε2 from (3.9) into (3.19), we obtain the following 12th order equation for ISCO radius rISCO:

R(r) =12∑n=0

cnrn = 0, (3.20)

where

c0 = a8m4(γ2 − 1)6,

c1 = −12a6m5(γ2 − 1)6,c2 = 6a

4m4(γ2 − 1)5[14m2(γ2 − 1)− a2(4γ2 − 1)

],

c3 = −2a2m3(γ2 − 1)4[144m4(γ2 − 1)2 + a4(13γ2 − 9)− 2a2m2(83γ4 − 70γ2 − 13)

],

c4 = 3m2(γ2 − 1)4

[− 2a6 + 192m6(γ2 − 1)2 − 8a2m4(−16− 27γ2 + 43γ4) + a4m2(−13 + 254γ2 + 50γ4)

],

c5 = 6m3(γ2 − 1)3

[16m4(γ2 − 1)2(22 + 9γ2) + a4(6 + 127γ2 + 53γ4)− 2a2m2(28− 256γ2 + 173γ4 + 55γ6)

],

c6 = m2(γ2 − 1)2

[4m4(γ2 − 1)2(844 + 648γ2 + 45γ4) + a4(101 + 424γ2 + 243γ4)− 2a2m2(504− 1965γ2 + 634γ4 + 791γ6 + 36γ8)

],

c7 = 6m(γ2 − 1)

[a4(9 + 22γ2 + 13γ4)− 2m4(γ2 − 1)2(−256− 268γ2 − 29γ4 + 9γ6)

+ a2m2(−141 + 396γ2 + 6γ4 − 236γ6 − 25γ8)],

c8 = 3[3a4(1 + γ2)2 +m4(γ2 − 1)2(580 + 704γ2 + 55γ4 − 78γ6 + 3γ8)− 4a2m2(29− 59γ2 − 27γ4 + 47γ6 + 10γ8)

],

9

c9 = −2m(1 + γ2)[a2(−36 + 42γ2 + 22γ4) +m2(314− 241γ2 − 181γ4 + 117γ6 − 9γ8)

],

c10 = −3(1 + γ2)[2a2(1 + γ2)−m2(47 + 3γ2 − 31γ4 + 5γ6)

],

c11 = 6m(γ2 − 3)(1 + γ2)2,

c12 = (1 + γ2)2. (3.21)

The equation R(r) = 0 generically has 12 solutions, which can be both real and complex. Physically, only two of theseroots should be real and positive, the larger radius corresponding to the retrograde orbit and the smaller one to theprograde orbit. We have verified numerically that it is indeed the case for a few arbitrarily chosen parameter values.Again, the Eq. (3.20) reduces to the Kerr case [39] when γ = 1.

3. The light ring

Null geodesics in the equatorial plane are described by a formalism essentially identical to the presentation abovefor massive particles, with the same Lagrangian (3.4) and conserved quantities (3.5). The only difference is that Eq.(3.14) gets replaced by

0 = gttṫ2 + 2gtφṫφ̇+ gφφφ̇

2 + grr ṙ2. (3.22)

The effective potential is then

ṙ2 = − 1grr

(gttṫ2 + 2gtφṫφ̇+ gφφφ̇

2) ≡ Veff . (3.23)

In terms of the conserved quantities, one has

Veff =1

grr∆(gttl

2 + 2gtφlε+ gφφε2) . (3.24)

We now restrict ourselves to circular orbits enforced by Veff = 0 and V′eff = 0, that is,

gttX2 + 2gtφX + gφφ = 0, g

′ttX

2 + 2g′tφX + g′φφ = 0, (3.25)

where we have defined the impact parameter X ≡ l/ε. The above equations give the value of X and the radius r ofthe null circular geodesic. An unstable circular orbit r = rc must also satisfy

V ′′eff(rc) > 0. (3.26)

Finally, we obtain an expression for both real and imaginary parts of QNMs:

Ωc =φ̇

ṫ

∣∣∣∣rc

=1

X(rc), and λ =

√V ′′eff2ṫ2

∣∣∣∣rc

. (3.27)

We will evaluate these expressions for each case studied in later sections.

To summarize, we have obtained the necessary results on geodesic motion in the KK metric. The angular momentumof a unit mass uncharged test particle moving along the ISCO of a KK BH is given by (3.13), where φ̇ and ε can becomputed using (3.9) and (3.11), and r = rISCO is the solutions of (3.20). The estimates for QNMs are provided by(3.27). The parameters m, a, and γ can be written in terms of the physical parameters M,Q, and A = J/M of theKK BH as

m =M

2

(3−

√1 + 2(

Q

M)2), (3.28)

a =

√2A(

1− ( QM )2 +√

1 + 2( QM )2

) 12

, (3.29)

γ2 =2 + ( QM )

2 + 2√

1 + 2( QM )2

4− ( QM )2. (3.30)

10

B. Pure geodesic final spin estimate

Armed with the above results on geodesic motion, we can estimate the final spin for a binary merger of KK BHs inthe Einstein-Maxwell-dilaton theory. Our estimate, strictly speaking, applies when one of the colliding BHs is neutral(since we ignored electromagnetic effects on the acceleration of the test particle, the charged case will be analyzed inthe following section); however, the computation is instructive to keep in mind even more generally since charges havemoderate effects on trajectories.

1. Bound on Af for the KK BHs

We can first derive an upper bound for the possible final spin generated by the merger. The metric (2.7) appearssingular when Σ = 0 and ∆ = 0. The former one is a curvature singularity, r = 0, θ = π/2, and the latter one is a

coordinate singularity, which turns out to consist of an inner horizon at r = m−√m2 − a2, and an event horizon at

r = m+√m2 − a2. In the standard interpretation of BH solutions, one imposes m2 ≥ a2 to avoid a naked singularity,

where the equality sign corresponds to the extremal limit.

In terms of the physical parameters M,Af , and Q, we arrive at the condition(3−

√1 + 2( QM )

2)(

1− ( QM )2 +

√1 + 2( QM )

2) 1

2

2√

2≥∣∣∣∣AfM

∣∣∣∣ , (3.31)where Q/M ∈ [0, 2). The allowed values of |Af/M | computed from (3.31) are shown in Fig. 1. According to theplot, the maximal spin |Af/M | for the KK BHs decreases as the charge to mass ratio Q/M increases. This generalobservation underlies our sense that the final spins of BH mergers can be lowered by introducing charges.

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

Q/M

|A f/M|

FIG. 1. The final spin |Af/M | vs. Q/M . The filled area illustrates possible values of Af/M , while the blue line represents theextremal value of |Af/M |. The final spin decreases as Q/M increases.

2. Final spin estimate for equal spin binary BH mergers

Consider initial BHs of equal spins χi = χ, the BKL formula (2.2) can then be rewritten as

Af = l(rISCO, Af )ν +M(1− 2ν)χ. (3.32)

Using the above equation, we can numerically solve for Af/M given ν and χ. First, consider the case χ = 0, i.e.,nonrotating binary BH coalescence, as shown in Fig. 2. When Q = 0, KK BHs reduce to Kerr BHs, and we get theusual GR value Af/M ' 0.66 for the equal mass case (ν = 0.25). When Q/M increases, Af/M obtained from theBKL recipe decreases.

11

0.00 0.05 0.10 0.15 0.20 0.250.0

0.1

0.2

0.3

0.4

0.5

0.6

ν

Af/M

Kerr BH

Q = 0.4 MQ = 0.8 MQ = 1.2 MQ = 1.6 M

FIG. 2. The final spin Af/M vs ν for χ = 0.

As another specific illustration, Fig. 3 shows the behavior of the final spin Af/M as a function of ν for χ = 0.4. Aswe can see from the plot, the final spin goes up from Af/M = 0.4 when ν rises from 0 to 0.25. Similar to the firstcase, Af/M decreases as Q/M increases.

0.00 0.05 0.10 0.15 0.20 0.250.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

ν

Af/M Kerr BHQ = 0.3 M

Q = 0.6 MQ = 0.9 M

FIG. 3. The final spin Af/M vs ν for χ = 0.4.

In contrast to the above two cases, for the nearly extreme spin parameters, say χ = 0.98, the value of the final spinAf/M falls while ν increases, as illustrated in Fig. 4.

0.00 0.05 0.10 0.15 0.20 0.25

0.945

0.950

0.955

0.960

0.965

0.970

0.975

0.980

ν

Af/M Kerr BH

Q = 0.1 MQ = 0.2 M

FIG. 4. The final spin Af/M plotted against varying ν and Q/M for the case χ = 0.98.

12

We remark that for extremely exotic values of the colliding BH parameters (large spins and charge-to-mass ratiosof order 1), the BKL estimate produces results that violate the maximal spin bound (3.31). This, of course, indicatesincompleteness of the recipe in the extreme regimes, but will not bother us here as we are only interested in moderatevalues of spins and charges.

IV. MERGER ESTIMATES FOR KK BHS FROM CHARGED PARTICLE MOTION

In the previous section, we only considered neutral test particles moving along geodesics of KK BHs. If individualcolliding BHs have charges, it is more natural to consider test particles subject also to electromagnetic interactions,which would make them deviate from pure geodesics. In this section, we will take into account the effect of theelectromagnetic field of the final BH on the motion of the test particle trajectories.

A. Kinematics

1. Circular orbits in the equatorial plane

We consider a test particle of mass µ and charge q moving around a charged rotating KK BH described by (2.7).The motion of the test particle follows from the Lagrangian

L = 12µgλν ẋ

λẋν − qAν ẋν . (4.1)

Because L does not depend on (t, φ), we have two conserved quantities, which are the energy and the angular momentumper mass of the test particle, respectively:

− ε = gttṫ+ gtφφ̇− eAt, l = gtφṫ+ gφφφ̇− eAφ, (4.2)

where we define e = q/µ.

Similar to Sec. III, we consider circular orbits in the equatorial plane, θ = π/2, θ̇ = 0, and ṙ = 0. The equations ofmotion are

gttṫ2 + 2gtφṫφ̇+ gφφφ̇

2 = −1, g′ttṫ2 + 2g′tφṫφ̇+ g′φφφ̇2 = 2e(A′tṫ+A′φφ̇). (4.3)

Using (4.2) and (4.3), we obtain the following expressions:

(ε− eAt)2 = φ̇2∆− gtt, l = −1

gtt

(φ̇∆ + gtφ(ε+ eAt)

)− eAφ. (4.4)

Combining (4.2) and (4.3), we get an equation determining φ̇,

b1φ̇4 + b2φ̇

3 + b3φ̇2 + b4φ̇+ b5 = 0, (4.5)

where the coefficients bi are functions of r defined by

b1 = 2g′φφ

(g′tt(2g2tφ − gttgφφ

)− 2gttgtφg′tφ

)+ gφφ

(4g′tφ

(gttg

′tφ − gtφg′tt

)+ gφφ (g

′tt)

2)

+ g2tt(g′φφ)

2

b2 = −4e(−gtφ

(A′t(gφφg

′tt + gttg

′φφ

)+ 2gttA

′φg′tφ

)+ gtt

(gφφ

(2A′tg

′tφ −A′φg′tt

)+ gttA

′φg′φφ

)+ 2g2tφA

′φg′tt

)b3 = 4e

2gtt(A′φ(gttA

′φ − 2gtφA′t

)+ gφφ (A

′t)

2)

+ 2g′tt(gφφg

′tt − 2gtφg′tφ

)+ gtt

(4(g′tφ)

2 − 2g′ttg′φφ)

b4 = 4e(gtt(A′φg

′tt − 2A′tg′tφ

)+ gtφA

′tg′tt

)b5 = 4e

2gtt (A′t)

2 + (g′tt)2 (4.6)

Note that if we neglect the electromagnetic influence on the motion, the odd powers of φ drop out, and we recover thesimple result (3.10).

We now turn to the ISCO radius. Using the normalization condition gµν ẋµẋν = −1 in the equatorial plane and Eqs.

(4.2), we arrive at the equation of motion

ṙ2 = Veff(r), (4.7)

13

where the effective potential is defined by

Veff =1

Br2(gtt(l + eAφ)

2 + 2gtφ(l + eAφ)(ε− eAt) + gφφ(ε− eAt)2 −∆), (4.8)

To find the ISCO radius, we have to impose the condition

d2

dr2Veff = 0 (4.9)

(in addition to enforcing the orbit to be circular).

2. The ISCO radius of a charged particle

Starting from the condition (4.9) and substituting l and ε computed from (4.2), we can solve numerically for theISCO radius. Figures 5 and 6 below show the values of rISCO plotted against e, the charge to mass ratio of the testparticle.

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.53.0

3.5

4.0

4.5

5.0

e

r ISCO/M Kerr BH

Q = 0.2 MQ = 0.4 MQ = 0.5 M

FIG. 5. ISCO radius rISCO/M vs e for A = 0.5M

.

With the electric charge assigned to the test particle, its angular momentum on circular orbits varies as the magnitudeof charge increases. As one can anticipate, the case eQ < 0 incurs an attractive force which helps to increase theangular momentum, while the case eQ > 0 produces the opposite effect as it incurs a repulsive force.

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.51.5

2.0

2.5

3.0

3.5

e

l(r ISCO)/M Kerr BH

Q = 0.2 MQ = 0.4 MQ = 0.5 M

FIG. 6. Angular momentum per mass l(rISCO)/M vs e for A = 0.5M .

14

B. Merger estimates for the KK BHs

1. Final spin

We are now able to perform the BKL estimation of the final spin according to (2.2), with electromagnetic effectstaken into account. We assume a positive final BH charge Q/M > 0 and vary the test particle charge e. As we haveexplained, the natural assignment for e in terms of the charges of colliding BHs is the ‘reduced charge’ Q1Q2/(Q1+Q2).The results for the case when both initial BHs are nonspinning (χ = 0) are shown in Fig. 7 (Q = 0.4M) and Fig. 8(Q = M).

0.00 0.05 0.10 0.15 0.20 0.250.0

0.1

0.2

0.3

0.4

0.5

0.6

ν

Af/M

0.16 0.18 0.20 0.22 0.240.45

0.50

0.55

0.60

0.65

Kerr BHe = -0.4e = -0.2e = 0e = 0.2e = 0.4

FIG. 7. The final spin Af/M vs ν for χ = 0, and Q = 0.4M .

0.00 0.05 0.10 0.15 0.20 0.250.0

0.1

0.2

0.3

0.4

0.5

0.6

ν

Af/M

Kerr BH

e = -1e = -0.5e = 0e = 0.2e = 0.4

FIG. 8. The final spin Af/M vs ν for χ = 0, and Q = M .

As visible from the plots, when the electromagnetic force between BHs enters into consideration, the final spin iscorrected. For BHs with opposite sign charges, the final spin is increased (and depending on the charge the value canbe larger from that resulting in Kerr binary BH collision). If the charges have the same sign, the final spin is smallerand is generally smaller than the one resulting in a Kerr collision. These behaviors also extend to the case when bothinitial BHs have nonzero spin (χ 6= 0), shown in Fig. 9. The resulting spin is always lower than for Kerr BHs.

15

0.00 0.05 0.10 0.15 0.20 0.25

0.4

0.5

0.6

0.7

ν

Af/M

Kerr BH

e = -1e = -0.5e = 0e = 0.2e = 0.4

FIG. 9. The final spin Af/M vs ν for χ = 0.4, and Q = 0.8M .

2. The light ring

For the Kaluza-Klein metric (2.7), we can compute the impact parameter X(r) by solving (3.25). It turns out that

X(r) =1

Ω(r)=−2maγ + rB

√∆

r − 2m. (4.10)

The radius of circular photon orbit is obtained by solving the equation

0 =r6 + 2(γ2 − 4

)mr5 +

(γ4 − 16γ2 + 24

)m2r4 − 2m

(a2(γ2 + 1

)+ 4

(γ4 − 5γ2 + 4

)m2)r3

− 2(γ2 − 1

)m2(a2(γ2 + 4

)− 8

(γ2 − 1

)m2)r2 − 8a2

(γ2 − 1

)2m3r + a4

(γ2 − 1

)2m2. (4.11)

We thus obtain the frequencies of QNMs (3.27) of the KK BHs. (We evidently recover the light ring of SchwarzschildBHs, r = 3m, by setting γ = 1 and a = 0.)

Figures 10 and 11 below show how the frequency parameters Ωc and λ change with initial BHs charges Q1, and Q2,in the equal-mass case with zero initial spins. We observe that the oscillation frequency Ωc decreases as the chargeratio increases, and the Lyapunov exponent λ increases. However, the differences are rather small; which is consistentwith the discussion of quasinormal modes in the case of nonspinning black holes in EMD theory [40].

• • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • •

•• • • • • • • • • • • • • • • • • • •

•• • • • • • • • • • • • • • • • • • •

-0.2 0.0 0.2 0.4 0.6 0.8 1.00.2700.275

0.280

0.285

0.290

0.295

Q2/Q1

Ω c×M

• • • • • • • • • • • • •• • • • • • •

• • • • • • • • • • • • •• • • • • • •

•• •

• • • •• • • • • • • • • • • • •

•• •

• • • •• • • • • • • • • • • • •

• •• • •

• • • • • • • • • • • •

• •• • •

• • • • • • • • • • • •

-0.2 0.0 0.2 0.4 0.6 0.8 1.00.1740.175

0.176

0.177

0.178

Q2/Q1

λ×M Q = 0.1 MQ = 0.2 MQ = 0.3 M

FIG. 10. Ωc and λ vs the charge ratio Q2/Q1 for different Q (KK BHs of equal masses with zero initial spins).

16

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.27

0.28

0.29

0.30

0.31

0.32

0.33

Q/M

Ω c×M

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.155

0.160

0.165

0.170

0.175

0.180

Q/M

λ×M

Q2 =Q1Q2 = 0.5 Q1Q2 = 0.2 Q1Q2 = 0Q2 = -0.1 Q1Q2 = -0.2 Q1Kerr BH

FIG. 11. Ωc and λ vs the total charge Q (KK BHs of equal masses with zero initial spins).

V. FINAL SPIN ESTIMATION FOR KERR-NEWMAN BHS

It is instructive to compare our above results for KK BHs in EMD gravity to their counterparts in ordinary gravity,the Kerr-Newmann BHs. Below, we essentially repeat the analysis of Secs. III-IV for the Kerr-Newman BHs describedby the metric (2.11).

A. Kinematics

In order to apply the BKL recipe, we need to find the angular momentum of a test particle orbiting the Kerr-Newman(KN) BH at the ISCO radius. For a test particle of mass µ and charge to mass ratio e, the energy per mass ε andangular momentum per mass l can be computed from the equations

(ε− eAt)2 = φ̇2∆− gtt, l = −1

gtt

(φ̇∆ + gtφ(ε+ eAt)

)− eAφ (5.1)

where now the metric and electromagnetic potential refer to the KN solution (2.11)-(2.13). The value of φ̇ can bedetermined from the equation

f1φ̇4 + f2φ̇

3 + f3φ̇2 + f4φ̇+ f5 = 0, (5.2)

with the coefficients

f1 =4(

4a2(Q2 −mr

)+(r(r − 3m) + 2Q2

)2)r2

f2 = −8aeQ(r −m)

r2

f3 = −4(Q2r

((e2 − 2

)r − 2

(e2 − 5

)m)

+(e2 − 4

)Q4 + 2mr2(r − 3m)

)r4

f4 =8aeQ

(Q2 −mr

)r5

f5 =8(e2 − 1

)mQ2r − 4

(e2 − 1

)Q4 + 4r2(m− eQ)(eQ+m)

r6. (5.3)

The ISCO orbit can be obtained by analyzing the effective potential Veff defined by

Veff =1

r2(gtt(l + eAφ)

2 + 2gtφ(l + eAφ)(ε− eAt) + gφφ(ε− eAt)2 −∆). (5.4)

17

The equation determining the ISCO radius rISCO is again of the form (4.9). If our test particle is taken to be neutral,the formulas simplify and yield explicitly

φ̇2 =

(Q2 −mr

) (3mr − 2Q2 − r2 ± 2a

√mr −Q2

)r2(

4a2 (Q2 −mr) + (r(r − 3m) + 2Q2)2) . (5.5)

where the upper sign is for prograde orbits, and the lower sign is for retrograde orbits. The ISCO radius is obtainedby solving the equation

a2(mr2(7m+ 3r) + 8Q4 − 2Q2r(7m+ 2r)

)+(mr2(6m− r) + 4Q4 − 9mQ2r

) (r(r − 3m) + 2Q2

)(5.6)

± 2a(4Q2 − 3mr

) (a2 + r(r − 2m) +Q2

)√mr −Q2 = 0.

Figures 12 and 13 show the values of ISCO radius rISCO, and angular momentum of the test particle at ISCO l(rISCO),plotted against e, the charge to mass ratio of the test particle.

-1.0 -0.5 0.0 0.5 1.0 1.53.0

3.5

4.0

4.5

5.0

e

r ISCO/M Kerr BHQ = 0.2 MQ = 0.4 MQ = 0.6 M

FIG. 12. ISCO radius rISCO/M vs e for a = 0.5M for Kerr-Newman BHs.

-1.0 -0.5 0.0 0.5 1.0 1.51.01.5

2.0

2.5

3.0

3.5

e

l(r ISCO)/M Kerr BH

Q = 0.2 MQ = 0.4 MQ = 0.6 M

FIG. 13. Angular momentum per mass l(rISCO)/M vs e for a = 0.5M for Kerr-Newman BHs.

18

B. Final spin of KN BHs coalescence

Consider the coalescence of two BHs with parameters (M1, Q1, A1) and (M2, Q2, A2), which results in a final BHwith parameters (M,Q,Af ), where M = M1 +M2 and Q = Q1 +Q2. The effect of electromagnetic fields on the finalspin Af can be seen from Fig. 14 below (where the initial spins are assumed to be zero).

0.00 0.05 0.10 0.15 0.20 0.250.0

0.1

0.2

0.3

0.4

0.5

0.6

ν

Af/M

Kerr BH

e = -0.6e = -0.2e = 0e = 0.1e = 0.6

FIG. 14. Kerr-Newman:The final spin Af/M vs ν for Q = 0.4M , and χ = 0.

Similarly to the KK case, final spins are lowered by the presence of charges (compared to Kerr collisions), exceptfor the situation with large charges of the opposite sign, which can make the final spin slightly higher than the Kerrcollisions. The situation is qualitatively similar for initially spinning BHs as shown in Fig. 15.

0.00 0.05 0.10 0.15 0.20 0.25

0.4

0.5

0.6

0.7

0.8

ν

Af/M

Kerr BH

e = -0.6e = -0.2e = 0e = 0.1e = 0.6

FIG. 15. Kerr-Newman:The final spin Af/M vs ν for Q = 0.4M , and χ = 0.4.

19

C. The light ring

Given the KN metric, we perform the light-ring analysis in the same manner as for the KK case. The impactparameter X(r) is given by

X(r) =1

Ω(r)=aQ2 − 2mar + r2

√∆

r2 − 2mr +Q2, (5.7)

and the radius of the null circular orbits can be obtained by solving

r4 − 6mr3 + (9m2 + 4Q2)r2 − 4m(a2 + 3Q2)r + 4Q2(a2 +Q2) = 0. (5.8)

As before, we can obtain from the solution an approximation to the oscillation frequency and decay rate of perturba-tions. The parameters Ωc and λ of the QNMs in this KN case display similar behavior to the KK case, as presentedin Figs. 16 and 17.

• • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • •

•• • • • • • • • • • • • • • • • • • •

•• • • • • • • • • • • • • • • • • • •

-0.2 0.0 0.2 0.4 0.6 0.8 1.00.2700.275

0.280

0.285

0.290

0.295

Q2/Q1

Ω c×M

• • • • • • • • • • • •• • • • • • • •

• • • • • • • • • • • •• • • • • • • •

•• •

• • • •• • • • • • • • • • • • •

•• •

• • • •• • • • • • • • • • • • •

•• •

• • •• • • • • • • • • • •

•• •

• • •• • • • • • • • • • •

-0.2 0.0 0.2 0.4 0.6 0.8 1.00.1740.175

0.176

0.177

0.178

Q2/Q1

λ×M Q = 0.1 MQ = 0.2 MQ = 0.3 M

FIG. 16. Kerr-Newman: Ωc and λ vs the charge ratio for various values of the total charge, in the equal-mass case and zeroinitial spins.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.27

0.28

0.29

0.30

0.31

0.32

0.33

0.34

Q/M

Ω c×M

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.155

0.160

0.165

0.170

0.175

0.180

Q/M

λ×M

Q2 =Q1Q2 = 0.5 Q1Q2 = 0.2 Q1Q2 = 0Q2 = -0.1 Q1Q2 = -0.2 Q1Kerr BH

FIG. 17. Kerr-Newman: Ωc and λ vs the total charge, in the equal-mass case and zero initial spins.

VI. COMPARISON BETWEEN KALUZA-KLEIN AND KERR-NEWMAN BHS

A. Final spins

For our comparison of KK BHs and KN BHs, we mainly restrict ourselves to the equal-mass case with zero initialspins (ν = 0.25 and χ = 0) and present the final spin as a function of the total charge Q = Q1 + Q2 and the initial

20

charge ratio Q2/Q1. We use the reduced charge assignment e = Q1Q2/Q for the test particle involved in the BKLestimate for both the KK and KN cases.

Figure 18 shows the values of the final spin for the KK case and KN case, which display qualitatively similarbehaviors. For fixed total charges, the final spins drop with the charge ratio. Additionally, depending on the chargeratios Q2/Q1, the final spin either increases or decreases with the total charge. The differences of final spins betweenKK case and KN case are presented in Fig. 19.

�������� • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • •

•••• • • • • • • • • • • • • • • • •

•••• • • • • • • • • • • • • • • • •

•

••••• • • • • • • • • • • • • • •

•

••••• • • • • • • • • • • • • • •

-0.2 0.0 0.2 0.4 0.6 0.8 1.00.60

0.62

0.64

0.66

0.68

0.70

0.72

0.74

Q2/Q1

Af/M • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • •

•• • • • • • • • • • • • • • • • • • •

•• • • • • • • • • • • • • • • • • • •

•••• • • • • • • • • • • • • • • • •

•••• • • • • • • • • • • • • • • • •

•

•

•••• • • • • • • • • • • • • • •

•

•

•••• • • • • • • • • • • • • • •

-0.2 0.0 0.2 0.4 0.6 0.8 1.00.60

0.62

0.64

0.66

0.68

0.70

0.72

0.74

Q2/Q1Af/M

Q = 0.1 MQ = 0.2 MQ = 0.3 MQ = 0.4 M

FIG. 18. Af/M vs Q2/Q1 for Kaluza-Klein (left) and Kerr-Newman (right) BHs.

• • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • •• • •

• • • • • • • • • • • • • • • • •• • •

• • • • • • • • • • • • • • • • •

••

• •• • •

• • • • • • • • • • • • •

••

• •• • •

• • • • • • • • • • • • •

-0.2 0.0 0.2 0.4 0.6 0.8 1.0-0.008-0.006-0.004-0.0020.000

0.002

0.004

Q2/Q1

(A f/M) KK-

(A f/M) KN

Q = 0.1 MQ = 0.2 MQ = 0.3 MQ = 0.4 M

FIG. 19. Difference of Af/M between Kaluza-Klein and Kerr-Newman BHs vs Q2/Q1

For equal charges Q1 = Q2, as presented in Fig. 20 for both the KN and KK cases, the final spin falls with increasingtotal charge. Additionally, at this value of Q2/Q1, the spins of KK BHs are greater than the spins of KN BH.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.50

0.55

0.60

0.65

Q/M

Af/M Kerr BH

Kerr-Newman BHKaluza-Klein BH

FIG. 20. Af/M vs Q/M (solid line) KN BHs, (dashed line) KK BHs, in the equal-charge case. (The value for Kerr BHs isindicated for reference).

21

It is also interesting to visualize the final spins for various charge ratios Q2/Q1, these are shown in Fig. 21. For lowcharge values, the predicted final spins for both KN and KK black holes are quite close but significant differences arisefor large charges.

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.62

0.63

0.64

0.65

0.66

0.67

0.68

Q/M

Af/M

Kerr BH

Q2 =Q1Q2 = 0.2 Q1Q2 = 0.05 Q1Q2 = 0Q2 = -0.05 Q1Q2 = -0.1 Q1Q2 = -0.15 Q1Q2 = -0.2 Q1

FIG. 21. Af/M vs Q/M (solid lines) KN BHs, (dashed line) KK BHs, for various charge ratios. (The value for the Kerr BH isindicated for reference).

Since the final spin value can be raised or lowered by electrically charged BHs, it is interesting to consider what initialspins (χ 6= 0) and charges give rise in an equal mass binary merger to a final BH with a spin parameter Af/M = 0.66.Figure 22 displays the necessary values of χ for both KK and KN cases. For the most ‘natural’ case of Q1 = Q2,depending on the charge, significantly spinning individual BHs are compatible with such a final outcome.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.0

0.1

0.2

0.3

0.4

Q/M

χ Q2 =Q1Q2 = 0Q2 = -0.1 Q1Q2 = -0.2 Q1

FIG. 22. χ vs Q/M that produce Af/M = 0.66. (solid lines) KN BHs, (dashed line) KK BHs, for various charge ratios.

B. The light ring

Figure 23 below presents Ωc and λ of both KK and KN BHs. We observe minor differences between these two cases.The real part Ωc of the KN case is smaller than the KK BHs, but this behavior reverses when the ratio Q2/Q1 reachesa certain value. In addition, the imaginary part λ of the KK BHs ia always bigger than for the KN BHs.

22

0.0 0.1 0.2 0.3 0.4 0.5 0.60.26

0.28

0.30

0.32

0.34

0.36

Q/M

Ω c×M

0.0 0.1 0.2 0.3 0.4 0.5 0.60.12

0.13

0.14

0.15

0.16

0.17

0.18

Q/M

λ×M

Q2 =Q1Q2 = 0.2 Q1Q2 = 0Q2 = -0.1 Q1Q2 = -0.2 Q1Kerr BH

FIG. 23. Fundamental frequencies of QNMs: (solid lines) KN BHs, (dashed line) KK BHs, for various charge ratios.

Figure 24 presents Ωc and λ compatible with the final spin Af/M =0.66 (from equal-mass binaries) of both KKand KN BHs.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.26

0.28

0.30

0.32

0.34

0.36

0.38

Q/M

Ω c⨯M

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.10

0.12

0.14

0.16

0.18

0.20

Q/M

λ⨯M Kaluza-Klein BHKerr-Newman BHKerr BH

FIG. 24. Fundamental frequencies of QNMs: (solid lines) KN BHs, (dashed line) KK BHs, for various total charges compatiblewith final spin of 0.66

VII. FINAL COMMENTS

In this work, we studied the main possible features of BH coalescence in Einstein-Maxwell-dilaton theory – for thespecific coupling value of α =

√3 –as well as the coalescence of charged BHs in GR. By applying straightforward

estimation techniques without adjustable parameters based on angular momentum conservation, we obtained approx-imate final spins of BH mergers. One particularly interesting observation drawn from our analysis is that the spin ofthe final BH is lowered when (an equal charge sign) BH coalescence is considered in our setup. This, in turn, impliesthat in such a merger, lower charged BHs will merge later than the more highly charged ones as the approximate“innermost stable circular orbit” lies at a lower frequency (larger radius) in the less charged BH case. Interestingly,we find that for both the KN and KK black holes merging with spins aligned with the orbital angular momentum, theeffect of individual charges in the black holes can contribute against the final black hole spin. In particular, for equalmass black holes (which in the nonspinning GR case yields a final spin with a value Af/Mf ' 0.66), a broad rangeof individual spin values can be compensated by suitable charges so as to provide the same final spin value. Sincethe effect of charges is subtle in the quasinormal frequencies, this observation highlights the importance of correlatingresults obtained during different stages of the merger (e.g. [4]) as well as digging deeper in the extraction of subleadingQNMs (e.g. [6–8])

23

The behavior hinted by the analysis presented here has been evidenced in fully nonlinear simulations [27], in asubclass of systems through a perturbative analysis [41] and through the analysis of geodesic motion in the KNgeometry [42]. As a final comment we stress that the strategy pursued here is applicable beyond the particulartheories we have focused on. Indeed, we expect that the same approach can be taken in any alternative gravity theory,once rotating BH solutions are known, and exploited to estimate the final BH parameters resulting from coalescenceand key quasinormal decay properties that can be confronted with observations.

ACKNOWLEDGMENTS

We thank David Chow, Thibault Damour, William East, Stephen Green, Eric Hirschman, Steve Liebling and CarlosPalenzuela for insightful discussions. This work was supported by a DPST Grant from the government of Thailand(to PJ); CUAASC grant from Chulalongkorn University (to AC and OE); NSERC through a Discovery Grant (to LL)and CIFAR (to LL). OE would like to thank Perimeter Institute for hospitality during the early stages of this work.Research at Perimeter Institute is supported through Industry Canada and by the Province of Ontario through theMinistry of Research & Innovation.

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I IntroductionII GeneralitiesA Final spin estimation: The BKL recipeB The light ringC Einstein-Maxwell-dilaton BHsD Newtonian limit of charged particle motion

III Merger estimates for KK BHs based on pure geodesic motionA Kinematic considerations1 Orbits in the equatorial plane2 The ISCO3 The light ring

B Pure geodesic final spin estimate1 Bound on Af for the KK BHs2 Final spin estimate for equal spin binary BH mergers

IV Merger estimates for KK BHs from charged particle motionA Kinematics1 Circular orbits in the equatorial plane2 The ISCO radius of a charged particle

B Merger estimates for the KK BHs1 Final spin2 The light ring

V Final spin estimation for Kerr-Newman BHsA KinematicsB Final spin of KN BHs coalescenceC The light ring

VI Comparison between Kaluza-Klein and Kerr-Newman BHs A Final spinsB The light ring

VII Final comments Acknowledgments References